Average Error: 14.0 → 0.3
Time: 10.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -86718073306.1840972900390625:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{\sin B \cdot {F}^{2}}, -\mathsf{fma}\left(1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)\right)\\ \mathbf{elif}\;F \le 150840120.32874381542205810546875:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -1 \cdot \frac{x \cdot \cos B}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B} + \frac{x}{\sin B \cdot {F}^{2}}, \frac{1}{\sin B}\right)\\ \end{array}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}
↓
\begin{array}{l}
\mathbf{if}\;F \le -86718073306.1840972900390625:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{\sin B \cdot {F}^{2}}, -\mathsf{fma}\left(1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)\right)\\

\mathbf{elif}\;F \le 150840120.32874381542205810546875:\\
\;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -1 \cdot \frac{x \cdot \cos B}{\sin B}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B} + \frac{x}{\sin B \cdot {F}^{2}}, \frac{1}{\sin B}\right)\\

\end{array}
double f(double F, double B, double x) {
        double r30243 = x;
        double r30244 = 1.0;
        double r30245 = B;
        double r30246 = tan(r30245);
        double r30247 = r30244 / r30246;
        double r30248 = r30243 * r30247;
        double r30249 = -r30248;
        double r30250 = F;
        double r30251 = sin(r30245);
        double r30252 = r30250 / r30251;
        double r30253 = r30250 * r30250;
        double r30254 = 2.0;
        double r30255 = r30253 + r30254;
        double r30256 = r30254 * r30243;
        double r30257 = r30255 + r30256;
        double r30258 = r30244 / r30254;
        double r30259 = -r30258;
        double r30260 = pow(r30257, r30259);
        double r30261 = r30252 * r30260;
        double r30262 = r30249 + r30261;
        return r30262;
}
↓
double f(double F, double B, double x) {
        double r30263 = F;
        double r30264 = -86718073306.1841;
        bool r30265 = r30263 <= r30264;
        double r30266 = 1.0;
        double r30267 = x;
        double r30268 = B;
        double r30269 = sin(r30268);
        double r30270 = 2.0;
        double r30271 = pow(r30263, r30270);
        double r30272 = r30269 * r30271;
        double r30273 = r30267 / r30272;
        double r30274 = cos(r30268);
        double r30275 = r30267 * r30274;
        double r30276 = r30275 / r30269;
        double r30277 = 1.0;
        double r30278 = r30277 / r30269;
        double r30279 = fma(r30266, r30276, r30278);
        double r30280 = -r30279;
        double r30281 = fma(r30266, r30273, r30280);
        double r30282 = 150840120.32874382;
        bool r30283 = r30263 <= r30282;
        double r30284 = r30263 / r30269;
        double r30285 = r30263 * r30263;
        double r30286 = 2.0;
        double r30287 = r30285 + r30286;
        double r30288 = r30286 * r30267;
        double r30289 = r30287 + r30288;
        double r30290 = r30266 / r30286;
        double r30291 = -r30290;
        double r30292 = pow(r30289, r30291);
        double r30293 = r30266 * r30276;
        double r30294 = -r30293;
        double r30295 = fma(r30284, r30292, r30294);
        double r30296 = -r30266;
        double r30297 = r30276 + r30273;
        double r30298 = fma(r30296, r30297, r30278);
        double r30299 = r30283 ? r30295 : r30298;
        double r30300 = r30265 ? r30281 : r30299;
        return r30300;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -86718073306.1841

    1. Initial program 26.9

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

    2. Simplified26.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)}\]

    3. Taylor expanded around -inf 0.2

      \[\leadsto \color{blue}{1 \cdot \frac{x}{\sin B \cdot {F}^{2}} - \left(1 \cdot \frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}\right)}\]

    4. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{x}{\sin B \cdot {F}^{2}}, -\mathsf{fma}\left(1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)\right)}\]

    if -86718073306.1841 < F < 150840120.32874382

    1. Initial program 0.4

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

    2. Simplified0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)}\]
    3. Using strategy rm

    4. Applied associate-*r/0.3

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -\color{blue}{\frac{x \cdot 1}{\tan B}}\right)\]

    5. Taylor expanded around inf 0.3

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -\color{blue}{1 \cdot \frac{x \cdot \cos B}{\sin B}}\right)\]

    if 150840120.32874382 < F

    1. Initial program 24.1

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

    2. Simplified24.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)}\]

    3. Taylor expanded around inf 0.2

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \left(1 \cdot \frac{x \cdot \cos B}{\sin B} + 1 \cdot \frac{x}{\sin B \cdot {F}^{2}}\right)}\]

    4. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B} + \frac{x}{\sin B \cdot {F}^{2}}, \frac{1}{\sin B}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -86718073306.1840972900390625:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{\sin B \cdot {F}^{2}}, -\mathsf{fma}\left(1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)\right)\\ \mathbf{elif}\;F \le 150840120.32874381542205810546875:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -1 \cdot \frac{x \cdot \cos B}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B} + \frac{x}{\sin B \cdot {F}^{2}}, \frac{1}{\sin B}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))